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With increasing requirements for greater precision, it becomes essential todescribe the effects of isospin breaking induced by both quark masses andelectro-magnetic effects. In this work we perform a lattice analysis of theweak decay constants of the neutral pseudoscalar mesons including such isospinbreaking effects, with particular consideration being given to the state mixingof the $\pi^0$, $\eta$ and $\eta^\prime$. We also detail extensions to thenon-perturbative RI$^\prime$-MOM renormalization scheme for application tonon-degenerate flavour-neutral operators which are permitted to mix, andpresent initial results. Using flavour-breaking expansions in terms of quarkmasses and charges we determine the leptonic decay constants for the $\pi^0$and $\eta$ mesons, demonstrating in principle how precision determinations ofall neutral pseudoscalar decay constants could be reached in lattice QCD withQED and strong isospin-breaking accounted for.

Weak decay constants of the neutral pseudoscalar mesons from lattice QCD+QED